Polar coordinates and equations

Polar coordinates and equations

• Polar coordinates involve the use of a radius and an angle to specify a point in a plane. The coordinates are written as (r, θ) where r is the distance from the origin and θ is the angle made with the positive x-axis.

• The basic formula to convert from Cartesian to Polar coordinates is r = √(x²+y²) and θ = tan⁻¹(y/x).

• Additionally, to convert from Polar to Cartesian coordinates, the formulas x = r cos θ and y = r sin θ are used.

• In polar coordinates, the equation of a line becomes r cos(θ - α) = p, where p is the perpendicular distance from the origin to the line, and α is the angle that the normal to the line makes with the positive x-axis.

• The equation of a circle with centre at the origin and radius a in polar coordinates is simply r = a.

• For a circle not centered at the origin, the equation takes the form r = a ± b cos(θ - α), or r = a ± b sin(θ - α), depending on the information given.

• When dealing with polar equations, it’s crucial to remember that r can be negative. This corresponds to a point that is in the opposite direction from the angle θ.

• The graph of a polar equation is known as a polar plot. When plotting, some polar equations can lead to special curves like limaçons or roses.

• Differentiation and integration with polar coordinates often involve multiplying by r. For instance, to find the area enclosed by a polar curve, the formula 1/2 ∫r² dθ is used.

• When working with polar coordinates, it is important to check for any values of θ for which the original function is not defined.